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Brazil Undergrad MO
2023 Brazil Undergrad MO
2
find the sum of the inverses of the central binomial coefficients
find the sum of the inverses of the central binomial coefficients
Source: Brazilian Mathematical Olympiad 2023, Level U, Problem 2
October 21, 2023
function
real analysis
Convergence
binomial coefficients
Problem Statement
Let
a
n
=
1
(
2
n
n
)
,
∀
n
≤
1
a_n = \frac{1}{\binom{2n}{n}}, \forall n \leq 1
a
n
=
(
n
2
n
)
1
,
∀
n
≤
1
.a) Show that
∑
n
=
1
+
∞
a
n
x
n
\sum\limits_{n=1}^{+\infty}a_nx^n
n
=
1
∑
+
∞
a
n
x
n
converges for all
x
∈
(
−
4
,
4
)
x \in (-4, 4)
x
∈
(
−
4
,
4
)
and that the function
f
(
x
)
=
∑
n
=
1
+
∞
a
n
x
n
f(x) = \sum\limits_{n=1}^{+\infty}a_nx^n
f
(
x
)
=
n
=
1
∑
+
∞
a
n
x
n
satisfies the differential equation
x
(
x
−
4
)
f
′
(
x
)
+
(
x
+
2
)
f
(
x
)
=
−
x
x(x - 4)f'(x) + (x + 2)f(x) = -x
x
(
x
−
4
)
f
′
(
x
)
+
(
x
+
2
)
f
(
x
)
=
−
x
.b) Prove that
∑
n
=
1
+
∞
1
(
2
n
n
)
=
1
3
+
2
π
3
27
\sum\limits_{n=1}^{+\infty}\frac{1}{\binom{2n}{n}} = \frac{1}{3} + \frac{2\pi\sqrt{3}}{27}
n
=
1
∑
+
∞
(
n
2
n
)
1
=
3
1
+
27
2
π
3
.
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